Stability of systems of linear equations and inequalities: distance to ill-posedness and metric regularity

Selected Applications of Linear Semi-Infinite Systems Theory

“…x 2 R 2 : 2x 1 + 2ux 2 < u 2 + 1; u 2 R is the set of Slater points of : Regarding the facial structure of the cells, to apply Theorem 4 or Corollary 6, one needs either the analyticity or the ful…llment or the LOP quali-…cation of their linear representations given by ( 10) and (11).…”

“…The chapters devoted to LSIS in the monograph [46] on LSIP describe the state-of-the-art on LSIS at the end of the 20th Century. The few papers on deterministic LSIS published since 1998 deal with the representation of closed convex sets by means of particular types of LSIS ( [1], [40], [60], [37]), while the many contributions to uncertain LSIS in this period cover a variety of topics, as the continuity properties of the solution set and associated maps ( [14], [51], [42], [43]), formulas involving error bounds and di¤erent types of distances and Lipschitz-like moduli ( [13], [10], [15], [12], [11], [16], [17], [18], [61]), as well as radius of robust feasibility [38].…”

Stability of the Feasible Set Mapping of Linear Systems with an Exact Constraint Set

“…The recent paper [6] extends to linear systems such that 0 < |E| ≤ n results on the interiority of σ in Θ R n c and provides formulae for the distance from σ to ill-posedness. Many other alternative approaches to the stability of the feasible set mapping are possible, e.g., the study of the topological behavior of the feasible set in the proximity of the nominal system (see, e.g., [15] and [16]), regularity properties and error bounds (see, e.g., [17] and [18]), etc.…”